You have connected the two windings in parallel, but there are two ways to do that. The correct way they are both winding around the transformer the same direction. This would be called in-phase. The wrong way is where the windings are in opposite directions, aka out-of-phase (by 180 degrees). In the later case, the magnetic fields generated by the two coils cancels, and it operates just like you short-circuited the mains. That would give a result similar to what you describe, except that I would expect it to immediately blow the circuit breaker while doing possibly catastrophic damage to the transformer, not sit there smoking. I guess it depends on how well coupled the two primary coils are. Perhaps there is enough leakage inductance to keep the current to several amps. A shorted turn or secondary winding might cause the behavior you describe, but that would be odd to say the least.
The earth ground shouldn't do anything to prevent a fault, it is only for safety after the fault happens. If the ground lead is carrying appreciable current it is a serious problem. Obviously the lack of a fuse is a problem as it would have limited the fault current to a safer value. On the other hand I have no explanation for how changing that could make it work better, even if you still doubt that it is really operating properly.
Transformers do heat up even when running at no load due to core losses. It is even conceivable that you could get the right voltage out at zero load, as it doesn't take much current to operate the regulator. If there is some short circuit or incorrect wiring, the voltage would collapse very quickly under load, so that would be easy to test with a suitable pair of power resistors.